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Modern Mathematics: An Introduction to MSRI's 2008-09 Programs October 10, 2007 - October 11, 2007

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Oct 10, 2007
Wednesday

04:00 PM - 04:30 PM		Welcome to MSRI
Location -- Video -- Abstract -- Supplements -- Show Detail
04:30 PM - 05:30 PM		Finding Rational Curves through Prescribed Points Brendan Hassett (Brown University)
Location -- Video -- Abstract It is a postulate of Euclidean geometry that through any two points in the plane there passes a unique line. A theorem of high-school geometry asserts the existence of a circle through any three non-collinear points. Lines and circles are instances of rational curves, and one central problem of modern algebraic geometry is to understand the rational curves passing through a given configuration of points in a surface (or higher-dimensional space). We review recent results and open problems that will be addressed at the MSRI program in Spring 2009. Supplements -- Show Detail
05:30 PM - 06:30 PM		Index formulas: a bridge between Topology, Geometry, Analysis, and Linear Algebra. Paul Loya
Location -- Video -- Abstract A fascinating feature of the Gauss-Bonnet formula is that it relates two invariants of a surface that are apparently distinct in nature, the Euler characteristic, which is topological in nature, and the total curvature, which is geometric in nature. The Atiyah-Singer index formula, proved in 1963 for closed manifolds, is a higher dimensional analog of the Gauss-Bonnet formula that incorporates a functional analytic invariant into the formula called the index of an operator. This work helped Atiyah and Singer win the Abel prize in 2004. In 1975, the Atiyah-Singer formula was extended to manifolds with boundary and in the late 90's it was extended to manifolds that have corners, in some sense the easiest class of "singular spaces." In the lecture, I will discuss the Atiyah-Singer formula from the viewpoint of the Gauss-Bonnet formula, and then talk about some recent results in extending the Atiyah-Singer formula to manifolds that have corners. Supplements -- Show Detail
07:00 PM - 09:00 PM		Dinner
Location -- Video -- Abstract -- Supplements -- Show Detail

Oct 11, 2007
Thursday

01:00 PM - 02:00 PM		Undergraduate Colloquium:Easy-to-explain but hard-to-solve problems in Convex Geometry Jesus De Loera (University of California, Davis)
Location -- Video -- Abstract Examples of convex sets are balls, cubes, triangles. These are figures that do not have "holes" or "valleys". Unknown to most people convex bodies are finding more and more applications in such diverse fields as optimization, statistics, algebra, or computer science. In this talk we will convince the audience that there is life after calculus and that even the most seasoned of mathematicians can't solve easy questions about convex bodies. Supplements -- Show Detail
02:00 PM - 02:30 PM		Introduction to MSRI
Location -- Video -- Abstract -- Supplements -- Show Detail
02:00 PM - 05:00 PM		Undergraduate mini-course on the Calculus of Analytic Critical Points Brendan Hassett (Brown University)
Location -- Video -- Abstract Every calculus student learns the second-derivative test for critical points: Depending on the sign of the second derivative, we can decide whether the critical point is a maximum or minimum. However, the test is inconclusive if the second derivative is zero. For functions in two variables, there is a second-derivative test for critical points using the matrix of second-order partial derivatives. This works well when the determinant of this matrix is nonzero; these are called nondegenerate critical points. Over the last three years, undergraduate students at Rice University have been doing research on critical points of analytic functions in two variables, classifying degenerate critical points using invariants generalizing those from multivariable calculus. Their work revolves around developing computational techniques for evaluating these invariants efficiently. We will present these techniques, as well as the mathematical principles underlying them. Supplements -- Show Detail
02:30 PM - 03:00 PM		Break
Location -- Video -- Abstract -- Supplements -- Show Detail
03:00 PM - 04:00 PM		Most Hard Equations are Easy J. Maurice Rojas (Texas A & M University)
Location -- Video -- Abstract While randomization has long been used profitably in optimization and computer science, it's importance in algebraic geometry has only recently been realized. In particular, the study of random systems of equations leads to some beautiful interactions between geometry, applications, and algorithmic complexity. We survey some of these developments from the point of view of polynomial system solving, focusing particularly on real (as opposed to complex) solutions. We will also see how centuries-old questions on counting real solutions can be solved with modern ideas from algebraic geometry (and a little randomization). No background in algebraic geometry or algorithms is assumed. Supplements -- Show Detail
04:00 PM - 05:00 PM		Ergodic Ramsey Theory Vitaly Bergelson
Location -- Video -- Abstract The main focus of this introductory lecture is the fascinating interplay between ergodic theory, Ramsey theory and Diophantine analysis. Ergodic theory has its roots in statistical and celestial mechanics and studies such phenomena as recurrence and uniform distribution of orbits. Ramsey theory, a branch of combinatorics, is concerned with the phenomenon of preservation of highly organized structures under finite partitions. Diophantine analysis concerns itself with integer and rational solutions of systems of polynomial equations. We will start with some examples which demonstrate the usefulness of the ergodic approach to combinatorics and number theory. The discussion will naturally lead us to some fascinating recent developments such as the celebrated Green-Tao theorem on arithmetic progressions in primes. We will conclude by formulating and discussing some natural open problems. Supplements -- Show Detail

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